|
%I #36 Sep 04 2023 16:39:16
%S 6,11,18,27,41,74,157,197,294,549,581
%N Exponents e such that the aliquot sequence starting with 2^e ends with a prime number at index 2.
%C That is, exponents e such that s(s(2^e)) is prime, where s(n) = sigma(n)-n (A001065).
%C Note that exponents e such that aliquot sequences starting with 2^e end with a prime number at index 1 (exponents e such that s(2^e) is prime) are called "Mersenne exponents" (see A000043).
%C From _Amiram Eldar_, Sep 02 2023:
%C Numbers k such that 2^k - 1 is a term of A037020.
%C 1206 < a(12) <= 2351 (2351 is a term). (End)
%H Jean-Luc Garambois, <a href="http://www.aliquotes.com/aliquotes_puissances_entieres/aliquotes_puissances_entieres.html">Aliquot sequences starting on integer powers n^i</a>.
%H Mersenne forum, <a href="https://www.mersenneforum.org/showpost.php?p=637222&postcount=2427">Results presentation page</a>.
%t Select[Range[100], PrimeQ[DivisorSigma[1, 2^# - 1] - 2^# + 1] &] (* _Amiram Eldar_, Sep 02 2023 *)
%o (Sage)
%o def s(n):
%o sn = sigma(n) - n
%o return sn
%o e = 1
%o exponents_list = []
%o while e<=200:
%o m = 2^e
%o index = 0
%o if is_prime(s(s(m))):
%o exponents_list.append(e)
%o e+=1
%o print (exponents_list)
%o (PARI) f(n) = sigma(n) - n; \\ A001065
%o isok(k) = ispseudoprime(f(f(2^k))); \\ _Michel Marcus_, Sep 02 2023
%Y Cf. A000043 (Mersenne exponents), A001065, A037020.
%K nonn,hard,more
%O 1,1
%A _Jean Luc Garambois_, Sep 02 2023
| 